Graph Signal Reconstruction Under Heterogeneous Noise via Adaptive Uncertainty-Aware Sampling and Soft Classification
Alessio Fascista, Angelo Coluccia, Chiara Ravazzi · IEEE Transactions on Signal and Information Processing over Networks · 2024
Reconstructing bandlimited graph signals from a subset of noisy measurements is a fundamental challenge within the realm of signal processing. Historically, this problem has been approached under the assumption of consistent noise variance throughout the entire network. Nevertheless, practical scenarios often present a diverse and heterogeneous noise landscape, greatly complicating the process of signal reconstruction. In this study, the task of reconstructing graph signals across networks where measurements may be affected by heterogeneous noise is addressed. A Bayesian model tailored for graph signals is considered, which takes into consideration the potential existence of node-specific variations in measurement variance, i.e., a different (and unknown) level of uncertainty. Moreover, a novel uncertainty-aware local graph coherence metric is introduced, which capitalizes on estimated parameters to refine the sampling process. By accommodating uncertainty, the accuracy of signal reconstruction is enhanced, even in the presence of demanding noise conditions. The proposed approach revolves around a unified framework that combines maximum likelihood and maximum a-posteriori principles. Specifically, the importance of each observation is weighted based on a soft classification of nodes, so incorporating the reliability of measurements into the reconstruction process. The latter is performed through a novel algorithm that couples re-weighted iterative least squares with expectation-maximization. Such an algorithm can effectively manage the varying noise levels and features a non-local total variation regularization term, which serves a dual purpose: it promotes sparsity in the reconstructed signal while preserving signal discontinuities, crucial for capturing the characteristics of the underlying graph signal. Extensive simulations demonstrate the effectiveness of the proposed approach for various graph topologies and anomalous conditions, revealing substantial enhancements in signal reconstruction accuracy compared to existing methods. An illustrative example on experimental PM10 data from the European Copernicus Atmosphere Monitoring Service (CAMS) is also reported.